What would be the time complexity If I will apply merge sort on an already sorted array?
Usual merge sort still uses O(nlogn) for sorted data.
But there is natural merge sort variant that provides linear complexity for sorted arrays.
Note that natural merge sort also gives O(nlogn) for arbitrary data, compared with isertion sort, that behaves well for sorted data but becomes quadratic in the worst case
According to the Wikipedia page for merge sort, merge sort has both a best and worse case performance of O(n log n). Given an input of an array already sorted, merge sort would still need to go through the same sorting process as for any other array. As a result, even for a sorted array, the running time would still be O(n log n).
For the case of an already-sorted array, there are other algorithms which actually beat merge sort, e.g. insertion sort. For insertion sort, the performance of an already sorted array is O(n), i.e. linear.
Related
Given n numbers that all are identical, then what would be the running time of merge sort?
Will it be in linear time O(n) or,
best case O(nlogn)
For a pure merge sort, the number of moves is always the same O(n log(n)). If all elements are the same or in order or reverse order, the number of compares is about half the number of compares for the worst case.
A natural merge sort that creates runs based on existing ordering of data would take O(n) time for all identical values or in order or reverse order. A variation of this is a hybrid insertion sort + merge sort called Timsort.
https://en.wikipedia.org/wiki/Timsort
You need to recheck the recursive formula that you have for the merge sort:
T(n) = 2T(n/2) + \Theta(n)
Now, when all values are identical, let see what will be changed in the formulation. \Theta(n) is for merging two subarrays. As the merging of two subarrays with identical members sweeps those arrays, independent of the identical members, it will be the same in your case.
Therefore, the recursion formula will be unchanged for the specified case; hence the time complexity will be Theta(n log n). That can be considered as one of the shortcomings of the mergesort.
Question;
You have to sort an array of bank transactions by date. Most of them
are in order (by date), only a few are out of order.
Which sorting algorithm will you use between insertion sort, selection
sort and merge sort in order to take advantage of the fact that the
array is almost sorted?
My answer (not sure if its correct)
Assuming that N >= 5, i would go with Merge Sort since its avg. time complexity would be O(n * log n) which would be more efficient than insertion sort O(n^2). However, since multiple transactions will be on the same dates, insertion sort would be a good STABLE sorting method.
Which one is better in this case? Merge or insertion sort? Am i in the right direction?
You should pick insertion sort, not because of stability, but because it's adaptive (see here) and will outperform due to the fact the input is almost sorted to begin with.
The meaning of this "adaptive" characteristic is that elements that are already in place are processed at O(1) time, and elements very close to their sorted position can also be considered O(1) (up to some k distance).
Quick sort has worst case time complexity as O(n^2) while others like heap sort and merge sort has worst case time complexity as O(n log n) ..still quick sort is considered as more fast...Why?
On a side note, if sorting an array of integers, then counting / radix sort is fastest.
In general, merge sort does more moves but fewer compares than quick sort. The typical implementation of merge sort uses a temp array of the same size as the original array, or 1/2 the size (sort 2nd half into second half, sort first half into temp array, merge temp array + 2nd half into original array), so it needs more space than quick sort which optimally only needs log2(n) levels of nesting, and to avoid worst case nesting, a nesting check may be used and quick sort changed to heap sort, (this is called introsort).
If the compare overhead is greater than the move overhead, then merge sort is faster. A common example where compares take longer than moves would be sorting an array of pointers to strings. Only the (4 or 8 byte) pointers are moved, while the strings may be significantly larger (and similar for a large number of strings).
If there is significant pre-ordering of the data to be sorted, then timsort (fixed sized runs) or a "natural" merge sort (variable sized runs) will be faster.
While it is true that quicksort has worst case time complexity of O(n^2), as long as the quicksort implementation properly randomizes the input, its average case (expected) running time is O(n log n).
Additionally, the constant factors hidden by the asymptotic notation that do matter in practice are pretty small as compared to other popular choices such as merge sort. Thus, in expectation, quicksort will outperform other O(n log n) comparison sorts despite the less savory worst case bounds
Not exactly like that. Quicksort is the best in most cases, however it's pesimistic time complexity can be O(n^2), it doesn't mean it always is. The issue lies in choosing the right point of pivot, if you choose it correctly you have time complexity O(n log n).
In addition, quicksort is one of the cheapest/easiest in implementation.
So I know the O(N) for linear is n, insertion is n**2, binary is log(n) and merge is nlogn
So Merge Sort is the best search for large lists. Which of the above is the best for small lists i.e. how small? Thanks
You're mixing up sort and search algorithms. Linear search and binary search are algorithms for finding a value in an array, not sorting the array. Insertion sort and mergesort are sorting algorithms.
Insertion sort tends run faster for small arrays. Many high-performance sorting routines, including Python's adaptive mergesort, automatically switch to insertion sort for small input sizes. The best size for the switch to occur is generally determined by testing. Java uses insertion sort for <= 6 elements in the primitive array versions of Arrays.sort; I'm not sure exactly how Python behaves.
You have got your facts wrong,
There is nothing called Linear Sort.
Insertion Sort is O(N^2)
There is nothing called Binary Sort
Though it could be Heap Sort which is O(NlogN)
MergeSort is O(NlogN)
QuickSoty is O(NlogN)
It is better to switch to insertion sort from merge sort if number of elements is less than 7
It is better to switch to insertion sort from quick sort if number of elements is less than 13
All of these sorting algorithms have an average case of O(n log n), so I would just like to know how I would be able to differentiate between these three sorting algorithms if I could run tests but not know which sorting algorithm was being run.
another difference between Heap and Merge sort you may want to concern is, Heap is not stable sort, but Mergesort is.
here is a table(link below), you could find (almost) any information about comparison sort algorithms you want.
https://en.wikipedia.org/wiki/Sorting_algorithm#Comparison_of_algorithms
heapsort is a inplace sorting algorithm , we don't need extra storage to sort the elements but mergesort is not inplace sorting algorithm , we required extra storage , in merge procedure , to sort the elements.The worst case running time of quicksort is O(n^2) that differentiate it form heapsort and mergesort
There are many cases in which performance of these algorithms are different.
For example.
if all input element are same.
then, heapsort will run in O(n) time
quicksort will run in O(n^2) time. (if last element is a pivote element)
and,
mergesort is going to take O(logn) time.